Arbelos

The term ``arbelos'' means Shoemaker's Knife in Greek, and this term is applied to the shaded Area in the above figure which resembles the blade of a knife used by ancient cobblers (Gardner 1979). Archimedes himself is believed to have been the first mathematician to study the mathematical properties of this figure. The position of the central notch is arbitrary and can be located anywhere along the Diameter.

The arbelos satisfies a number of unexpected identities (Gardner 1979).

1. Call the radii of the left and right Semicircles $a$ and $b$, respectively, with $a+b\equiv R$. Then the arc length along the bottom of the arbelos is

$$L=2\pi a+2\pi b=2\pi(a+b)=2\pi R,$$

so the arc lengths along the top and bottom of the arbelos are the same.

2. Draw the Perpendicular $BD$ from the tangent of the two Semicircles to the edge of the large Circle. Then the Area of the arbelos is the same as the Area of the Circle with Diameter $BD$.

3. The Circles $C_1$ and $C_2$ inscribed on each half of $BD$ on the arbelos (called Archimedes' Circles) each have Diameter $(AB)(BC)/(AC)$. Furthermore, the smallest Circumcircle of these two circles has an area equal to that of the arbelos.

4. The line tangent to the semicircles $AB$ and $BC$ contains the point $E$ and $F$ which lie on the lines $AD$ and $CD$, respectively. Furthermore, $BD$ and $EF$ bisect each other, and the points $B$, $D$, $E$, and $F$ are Concyclic.

5. In addition to the Archimedes' Circles $C_1$ and $C_2$ in the arbelos figure, there is a third circle $C_3$ called the Bankoff Circle which is congruent to these two.

6. Construct a chain of Tangent Circles starting with the Circle Tangent to the two small ones and large one. The centers of the Circles lie on an Ellipse, and the Diameter of the $n$th Circle $C_n$ is ($1/n$)th Perpendicular distance to the base of the Semicircle. This result is most easily proven using Inversion, but was known to Pappus, who referred to it as an ancient theorem (Hood 1961, Cadwell 1966, Gardner 1979, Bankoff 1981). If $r\equiv AB/AC$, then the radius of the $n$th circle in the Pappus Chain is

$$r_n={(1-r)r\over 2[n^2(1-r)^2+r]}.$$

This general result simplifies to $r_n=1/(6+n^2)$ for $r=2/3$ (Gardner 1979). Further special cases when $AC=1+AB$ are considered by Gaba (1940).

7. If $B$ divides $AC$ in the Golden Ratio $\phi$, then the circles in the chain satisfy a number of other special properties (Bankoff 1955).

See also Archimedes' Circles, Bankoff Circle, Coxeter's Loxodromic Sequence of Tangent Circles, Golden Ratio, Inversion, Pappus Chain, Steiner Chain

References

Bankoff, L. ``The Fibonacci Arbelos.'' Scripta Math. 20, 218, 1954.

Bankoff, L. ``The Golden Arbelos.'' Scripta Math. 21, 70-76, 1955.

Bankoff, L. ``Are the Twin Circles of Archimedes Really Twins?'' Math. Mag. 47, 214-218, 1974.

Bankoff, L. ``How Did Pappus Do It?'' In The Mathematical Gardner (Ed. D. Klarner). Boston, MA: Prindle, Weber, and Schmidt, pp. 112-118, 1981.

Bankoff, L. ``The Marvelous Arbelos.'' In The Lighter Side of Mathematics (Ed. R. K. Guy and R. E. Woodrow). Washington, DC: Math. Assoc. Amer., 1994.

Cadwell, J. H. Topics in Recreational Mathematics. Cambridge, England: Cambridge University Press, 1966.

Gaba, M. G. ``On a Generalization of the Arbelos.'' Amer. Math. Monthly 47, 19-24, 1940.

Gardner, M. ``Mathematical Games: The Diverse Pleasures of Circles that Are Tangent to One Another.'' Sci. Amer. 240, 18-28, Jan. 1979.

Heath, T. L. The Works of Archimedes with the Method of Archimedes. New York: Dover, 1953.

Hood, R. T. ``A Chain of Circles.'' Math. Teacher 54, 134-137, 1961.

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 116-117, 1929.

Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 54-55, 1990.